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Thermal transport from
first-principles DFT calculations
Keivan Esfarjani
Department of Mechanical Engineering
MIT
5/23/2012 Phonon School @ UWM 1
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Classical MD simulations …
… use an empirical potential fitted to
reproduce lattice constant, cohesive
energy, elastic properties, some
phonons…
But are not supposed to produce correct
third derivatives of the potential energy,
determining phonon lifetimes
They can be used to explain trends in
thermal transport, but can not predict
quantitatively the thermal conductivity. 5/23/2012 Phonon School @ UWM 2
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Goal
To develop a general methodology to get
accurate estimates of thermal properties
using available density functional theory
(DFT) tools.
To make accurate / reliable predictions
without fitting any parameters to
experimental data.
5/23/2012 Phonon School @ UWM 3
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Outline
New force field based on FP-DFT
Approaches for thermal conductivity
calculation + illustrations
◦ Lattice dynamics model
◦ MD
◦ Green’s functions for coherent transport
Conclusions
5/23/2012 Phonon School @ UWM 4
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Force field model
Derivatives are calculated at or near equilibrium
i i i ij j ijk j k
j, jk,
ijkl j k l
jkl,
2 3
i ij ijk
i i j i j k
4
ijkl
i j k l
1F E u u u
2!
1u u u
3!
E E E; ; ;
u u u u u u
E
u u u u
Force displacements
Exact for small displacements! 5/23/2012 Phonon School @ UWM 5
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They need to be reduced
As the FCs are 2nd or higher rank tensors,
there are a huge number of terms in the
Taylor expansion!
9 for each pair * number of pairs
27 for each triplet * number of triplets
81 for each quadruplet * number of quadruplets
5/23/2012 Phonon School @ UWM 6
Method to extract anharmonic force constants from first principles calculations
Keivan Esfarjani and Harold T. Stokes
Phys. Rev. B 77, 144112 (2008)
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Symmetry constraints
The FCs are related by the following
symmetry constraints:
Permutation of the order of derivatives
Global Translational invariance
jiij
ij ijk
j j
0 ; 0
5/23/2012 Phonon School @ UWM 7
Method to extract anharmonic force constants from first principles calculations
Keivan Esfarjani and Harold T. Stokes, Phys. Rev. B 77, 144112 (2008)
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Symmetry constraints
Global Rotational invariance Links FCs to higher-order ones
Group symmetry operation invariance
These symmetries must be enforced for any
FC model to be physically correct
R R
R ; R
' '
ij ' ' S(i)S( j)
' '
S S ...
5/23/2012 Phonon School @ UWM 8
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Methodology : FP-DFT calculations
in a supercell (real space)
Supercell size should ideally be larger than the
range of the force constants
5/23/2012 Phonon School @ UWM 9
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FU
0Invariance constraints
FP-DFT Force-displacement data
Extraction of FCs: • Constraints are linear in FCs • Force-displacement relations also linear in FCs
• Over-complete system of linear equations
• FCs are deduced from a SVD algorithm
5/23/2012 Phonon School @ UWM 10
i i ij j ijk j k
j, jk,
1F u u u ...
2!
Method to extract anharmonic force constants from first principles calculations
Keivan Esfarjani and Harold T. Stokes, Phys. Rev. B 77, 144112 (2008)
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Move one Si atom along [111]
How accurate are the obtained FCs?
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More validation
Use the FCs to calculate forces for
arbitrary atomic displacements
Denominator s(Poly) s(SW)
DISP=0.1 Ang 205.67 0.046941 0.34490
DISP=0.2 Ang 1166.9 0.082234 0.28324
Error is due to choosing (5,1,1) neighbors in the model 5/23/2012 Phonon School @ UWM 12
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Now what ?
Use second derivatives to get phonon
dispersion
Use the third derivatives to get phonon
lifetimes
Use all to do MD
Alternative approach to determining FCs uses
DFPT
(See talk by Derek Stewart)
5/23/2012 Phonon School @ UWM 13
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5/23/2012 14
FP-DFT in a supercell
Model
Potential
Thermal
Properties
MD-GK Latt
Dyn
Summary of the approach
Phonon School @ UWM
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ZrCoSb Half-heusler; By J. Shiomi, KE, G. Chen, PRB 84 (2011)
Phonon Dispersion
For Si, Graphene, ZrCoSb
5/23/2012 Phonon School @ UWM 15
kk
d ln
d ln V
N. Mingo, K. Esfarjani, D. A. Broido, and D. A. Stewart
Phys. Rev. B 81, 045408 (2010)
K. Esfarjani, G. Chen, and H. T. Stokes
Phys. Rev. B 84, 085204 (2011)
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RuleGolden Fermi)(||||2
),( 2
fi EEfVifiW
q1
q2
Normal process
qi
G
q2
q1
Umklapp process
qi
,,
...!4
1
!3
1
ijkl
lkjiijkl
ijk
kjiijk uuuuuuuV
Phonon creation and annihilation operators
q2
q1
a 4-phonon process
qi
q3
Lifetimes due to 3-phonon processes
5/23/2012 Phonon School @ UWM 16
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Thermal conductivity
Use of the relaxation time approximation
Cv per mode Relaxation time
5/23/2012 Phonon School @ UWM 17
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ZrCoSb lifetimes: LD-FGR versus MD
0
10
20
30
40
50
60
70
0 2 4 6 8 10 12 14 16
Frequency (Thz)
Velocity autocorrelation (5x5x5 supercell=1000 Si atoms)
5/23/2012 Phonon School @ UWM 18 Shiomi, Esfarjani, Chen, Phys. Rev. B 84, 104302 (2011)
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Thermal conductivity of ZrCoSb
Simulations vs Experiments
40/1/1 B
Impurity scattering
Y. Xia et al., J. Appl. Phys. 88, 1952 (2000).
T. Sekimoto et al., Jpn. J. Appl. Phys. 46, L673 (2007).
Co
Zr
Sb
Shiomi, Esfarjani, Chen, Phys. Rev. B 84, 104302 (2011)
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0 0.2 0.4 0
5
10
15
20
25
x
k (W
m -1
K -1
)
Hf x Zr
1-x CoSb
Hf 1-x
Ti x CoSb
Zr x Ti
1-x CoSb
Alloying effect on half-Heusler thermal conductivity
Experiment
Hf0.5Zr0.5CoSb0.8Sn0.2
(Yan, Nano Lett, 2011)
Transferability of force constants
Co
Zr
Sb
Co
Sb
Only changing mass
Hf
16 Wm-1K-1
22 Wm-1K-1
Green Kubo Calculation (300 K)
ZrCoSb
HfCoSb
Shiomi, Esfarjani, Chen, Phys. Rev. B 84, 104302 (2011)
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Thermal conductivity of PbTe
Takuma Shiga
Why is thermal conductivity
of PbTe so low
(~2 Wm-1K-1 @300K)?
Rocksalt PbTe
0
2
4
0
8
16
0
10
X LK
Fre
quen
cy (
TH
z)
This workW. Cochran, et al., (1966) Freq
uen
cy (m
eV)
Gru
nei
sen
par
amet
er
TO
Microscopic mechanism of low thermal conductivity in lead telluride
T Shiga, J Shiomi, J Ma, O Delaire, T Radzynski, A Lusakowski, K E, and G Chen
Phys. Rev. B 85, 155203 (2012)
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Phonons in lead telluride
Takuma Shiga
Microscopic mechanism of low thermal conductivity in lead telluride
T Shiga, J Shiomi, J Ma, O Delaire, T Radzynski, A Lusakowski, K E, and G Chen
Phys. Rev. B 85, 155203 (2012)
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5/23/2012 Phonon School @ UWM 23
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Summary of LD: MFP distribution
5/23/2012 Phonon School @ UWM 24
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MD using the polynomial force field
PROS:
◦ Of accuracy similar to DFT
◦ Includes anharmonic effects to higher order
CONS:
◦ Can not describe structural changes
◦ Hard to include surfaces and defects
Ideal for thermal conductivity using GK
5/23/2012 Phonon School @ UWM 25
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Equilibrium Green-Kubo (GK)
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Half-Heuslers
0 10 20 30 40 50 600
5
10
15
20
25
t (ps)
k (W
/m
K)
kxx
kyy
kzz
5/23/2012 Phonon School @ UWM 27
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Silicon
5/23/2012 Phonon School @ UWM 28
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Wave effects in thermal transport
Quantization of thermal conductance
Coherent (ballistic) phonon contribution
5/23/2012 Phonon School @ UWM 29
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Coherent phonon transport
Landauer formula
Transmission formula = 1
m gm
m 0
d n(T, )G D ( ) v ( )
2 T
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Experimental samples
5/23/2012 Phonon School @ UWM 31
12 nm GaAs
12 nm AlAs
7 periods
3 periods
~100 nm Al
350μm GaAs
12 nm GaAs 12 nm AlAs
1 period
5 periods
9 periods
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GaAs-[(AlAS)22/(GaAs)22]N=1,…,9 -Al
AlAs AlAs Al GaAs GaAs GaAs
1 unit= 22 layers AlAs+22 layers GaAs
Period length=2*(22*5.4 Ang)=23.76 nm
N-periods (N=1,3,5,7,9); 3x3 cross section
for a total of 1584,…,up to 14256 atoms
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(a) (b)
Transmission and Conductivity
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Conclusions
The developed force field can be calculated for an arbitrary
crystal structure
It can be used to:
◦ Do LD, get phonon spectra, lifetimes, k (fast)
◦ Do MD, get lifetimes and k including alloy scattering and anharmonic
effects nonperturbatively
◦ Set up the GFs to calculate transmission & conductance
LD faster and more accurate than GK-MD, but valid at low T
In nanostructured materials:
◦ Wave effects are important,
◦ Heat transport can have a large ballistic component
◦ Boundary scattering can be dominant
◦ Optical modes contribution is increased
5/23/2012 Phonon School @ UWM 34
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Acknowledgements
Gang Chen (MIT)
Junichiro Shiomi (U Tokyo) ZrCoSb, PbTe
Takuma Shiga (U Tokyo) PbTe
Maria Luckyanova (MIT) GaAs/AlAs SLs
J. Garg (MIT) GaAs/AlAs phonon lifetimes
TF Luo (Notre Dame) GaAs
5/23/2012 Phonon School @ UWM 35